Shearing Flow with Constant Stress

A rigid dumbbell suspension is at rest for t 0 in a space between two parallel planes perpendicular to the y-axis. For t O one of the planes moves in the positive x-direction in such a way that the shear stress has a constant value tq. For t 0 the strain y and the velocity gradient k are functions of time, and 

As t- cc, K will approach the constant value k clearly Zyx=—tiK where t] is the non-Newtonian viscosity given for rigid dumbbell suspensions by Eq. (6.7). 

The plot of y against t becomes linear as t becomes large, and the intercept of the linear portion of the curve on the y-axis will be called yo (see Fig. 8). From this definition the intercept is given by  

Another deflned quantity often referred to in tte polymer literature [Ferry (26), p. 15] is the equilibrium shear compliance J  

This equation is the same as Eq. (12.2) save for the inclusion of the term Jxq here. 

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Inception of Shearing Flow with Constant Stress ( Creep ) [Bird and Warner (9)]... 

In the inception of shearing flow with constant stress, the rigid dumbbells give Je dependent on kx, whereas the elastic dumbbells do not. 

Upon starting a steady shear flow with constant shear rate yQ = q the shear stress for a Maxwell element increases gradually in time to a steady state value rjq according to (see Chap. 13)... 

In the experiments, two main types of time-dependent flows have been studied start-up flows and stress relaxation. In the start-up flow experiments, shear flows with constant shear rates and elongational flows with constant elongational rates are started in the system in equilibrium under no external force, and the time-dependent stress build-up in the system is measured. In the stress relaxation experiments, constant deformations are applied to or removed from the system, and the time-dependent relaxation of the stress is measured. In this section, we study these two types within the framework of transient network theory. 

Exponential shear is therefore not a flow with constant stress history. The stress in this flow tends to grow without limit, even in an inelastic or linearly elastic fluid, and this makes the presentation of data an important issue. Doshi and Dealy [57] and Dealy [58] have argued that the results of an exponential shear experiment should be reported in terms of a time-dependent exponential viscosity rf that is defined in terms of the instantaneous shear rate ... 

With the gel equation, we can conveniently compute the consequences of the self-similar spectrum and later compare to experimental observations. The material behaves somehow in between a liquid and a solid. It does not qualify as solid since it cannot sustain a constant stress in the absence of motion. However, it is not acceptable as a liquid either, since it cannot reach a constant stress in shear flow at constant rate. We will examine the properties of the gel equation by modeling two selected shear flow examples. In shear flow, the Finger strain tensor reduces to a simple matrix with a shear component... 

Fig. 13. Shear stress t12 and first normal stress difference N1 during start-up of shear flow at constant rate, y0 = 0.5 s 1, for PDMS near the gel point [71]. The broken line with a slope of one is predicted by the gel equation for finite strain. The critical strain for network rupture is reached at the point at which the shear stress attains its maximum value...

When these, or other dimensionally correct, relaxation terms are added to the right sides of Eqs. (9-43) and (9-44), distinctive scaling relationships can be derived, which can be expressed as follows. Let a t, [y(t)]) be the stress tensor at time t during a shearing flow with shear history [y (f)]. Now choose a new shear history [y (t)] = c[y(cf)] in which the shear rate at each instant r is a constant c times the shear rate in the old history at time ct. Then, the stress a in the new shear history is given by... 

Vane rheometry provides a direct measurement of the shear stress at which flow is evident tmder the conditions of test (i.e. within the time scale of the measurement). In a constant rate (CR) experiment the material is sheared at a low but constant rate according to the speed of rotation of the viscometer s spindle, and the corresponding torque response with time is recorded. With constant stress (CS) experiments it is the inferred deformation that is recorded as a ftmction of time, imder the apphcation of a series of controlled and constant shear stresses. 

For start-up of shear flow at constant rate, the transient viscosity grows in a power law with time. This might be utilized for detecting GP. The total strain must be kept small because, near GP, stress relaxation is infinitely slow and shear modification cannot be avoided even at extremely low rates of deformation. 

The flow of a viscoelastic liquid between infinite parallel walls is a viscometric flow, or a flow with constant stretch history. The velocity profile for a fluid that is isotropic at rest is determined in such a flow only by the shear viscosity, although the stress distribution depends on the viscoelastic parameters. A nearly parallel flow for which the Deborah number is low, and stress growth and relaxation is not important, can be treated as though the local flow were that between infinite parallel walls in that case the viscoelasticity is not important for determining the flow field and the process can be analyzed with the lubrication or Hele-Shaw equations as though the poljmier were purely viscous. Effects attributable to the viscoelastic parameters (eg, interface movement in co-extrusion)... 

The purpose of this paper is to use the Flexible Wall Biaxial Tester to get more insight in powder flow behavior. This has been done by preparing powder samples and shearing them with constant volume and with eight different types of deformation. Anisotropy is occurring in these samples due to the structure in the powder. It was seen that stresses on opposite walls differ which means that there are shear stresses. It is thought that these are at least partially caused by the powder and not by the tester. This would mean that the principal axes of stress are not in the same direction as the principal axes of strain. 

The parallel plate system is a flow chamber that applies laminar shear flow with an incompressible fluid (i.e., homogeneous and has constant density throughout) by a pressure differential (Figure 15.4a), where the wall shear stress (xj is given by (Bacabac et al. 2005) ... 

Memory effects are revealed by experiments in which a complex fluid is subject to a time-dependent shear rate. Included under this rubric are measurements of the startup stress when a constant shear rate is suddenly imposed on an initially stationary system, and the stress when a system subject to some nonzero constant rate of strain suddenly has the rate of strain increased, decreased, or reversed. A prominent feature in measurements of stress on sudden imposition of a large rate of strain is stress overshoot, in which the stress first increases to a value much larger than its steady-state value, and then relaxes back to its steady-state value. Contrariwise, if the shear rate applied to a polymer fluid is held constant for a long time and then suddenly reduced, the stress may show undershoot the stress declines to a value well below its steady-state value and then increases back to its steady-state value. Related features have been seen for N. Bird, et al. also note measurements on responses to superposed flows, in particular the combination of a constant rate of shear flow with an oscillatory shear parallel or perpendicular to the constant shear(7). Bird, et al. further assert that multiple oscillations around the steady-state stress are sometimes observed before the steady state is attained. Recent studies involving step strains or oscillatory shear superposed on steady shear are reported by Li and Wang(8). 

Hence, it is found that a Kelvin solid flows under constant stress once time is of the order of Tr (Figure 6.17a). Some authors refer to Kelvin Tr as retardation time. As discussed below, glass delayed elasticity and flow can be captured with a Burger solid that combines in series a Kelvin and a Maxwell solid. A Kelvin solid yields retardation while a Maxwell one yields relaxation. Relaxation time informs on the time scales at which a viscoelastic solid will behave elastically or relax. Let us consider glass transition the viscosity is 10 -Pas while shear modulus of most glasses scales with Pa so that relaxation time is of the... 

The Weissenberg number is introduced in the following section in connection with flows with constant stretch history, i.e., flows in which the deformation rate and aU the stresses are constant with time. These are flows in which De is zero. And for deformations in which linear viscoelastic behavior is exhibited, Wi is zero. However, there are also flows of practical importance in which both Wi and De are nonzero and are sometimes even directly related to each other. This causes confusion, as authors often use the two groups interchangeably. This situation arises, for example, in the flow from a reservoir into a much smaller channel, either a slit or capillary. A Weissenberg number can readily be defined for this flow as the product of the characteristic time of the fluid and the shear rate at the wall of the flow channel. However, entrance flow is clearly not a flow with constant stretch history, and the Deborah number is thus non-zero as well and depends on the rate of convergence of the flow. 

Notice from Figure 8.12 that 0 varies from ti/2 - a to a for the cone-and-plate geometry. The shear stress difference across the gap is therefore Ax/A(r) = [sin (jt/2-a)] - 1 0 for small cone angles a. Therefore, provided that cones with small angles are used, the shear stress in a cone-and-plate shear flow is constant irrespective of location 0 in the gap. 

Consistent with this model, foams exhibit plug flow when forced through a channel or pipe. In the center of the channel the foam flows as a soHd plug, with a constant velocity. AH the shear flow occurs near the waHs, where the yield stress has been exceeded and the foam behaves like a viscous Hquid. At the waH, foams can exhibit waH sHp such that bubbles adjacent to the waH have nonzero velocity. The amount of waH sHp present has a significant influence on the overaH flow rate obtained for a given pressure gradient. 

These two moduli are not material constants and typical variations are shown in Fig. 5.3. As with the viscous components, the tensile modulus tends to be about three times the shear modulus at low stresses. Fig. 5.3 has been included here as an introduction to the type of behaviour which can be expected from a polymer melt as it flows. The methods used to obtain this data will be described later, when the effects of temperature and pressure will also be discussed. 

Viscosity is usually understood to mean Newtonian viscosity in which case the ratio of shearing stress to the shearing strain is constant. In non-Newtonian behavior, which is the usual case for plastics, the ratio varies with the shearing stress (Fig. 8-5). Such ratios are often called the apparent viscosities at the corresponding shearing stresses. Viscosity is measured in terms of flow in Pa s, with water as the base standard (value of 1.0). The higher the number, the less flow. 

A straightforward estimate of the maximum hardness increment can be made in terms of the strain associated with mixing Br and Cl ions. The fractional difference in the interionic distances in KC1 vs. KBr is about five percent (Pauling, 1960). The elastic constants of the pure crystals are similar, and average values are Cu = 37.5 GPa, C12 = 6 GPa, and C44 = 5.6 GPa. On the glide plane (110) the appropriate shear constant is C = (Cu - C12)/2 = 15.8 GPa. The increment in hardness shown in Figure 9.5 is 14 GPa. This corresponds to a shear flow stress of about 2.3 GPa. which is about 17 percent of the shear modulus, or about C l2n. 

The second category, time-dependent behaviour, is common but difficult to deal with. The best known type is the thixotropic fluid, the characteristic of which is that when sheared at a constant rate (or at a constant shear stress) the apparent viscosity decreases with the duration of shearing. Figure 1.21 shows the type of flow curve that is found. The apparent viscosity continues to fall during shearing so that if measurements are made for a series of increasing shear rates and then the series is reversed, a hysteresis loop is observed. On repeating the measurements, similar behaviour is seen but at lower values of shear stress because the apparent viscosity continues to fall. 

Whilst the flow curves of materials have received widespread consideration, with the development of many models, the same cannot be said of the temporal changes seen with constant shear rate or stress. Moreover we could argue that after the apparent complexity of linear viscoeleastic systems the non-linear models developed above are very poor cousins. However, it is possible to introduce a little more phenomenological rigour by starting with the Boltzmann superposition integral given in Chapter 4, Equation (4.60). This represents the stress at time t for an applied strain history ... 

Constant-Stress Layer in Flowing Fluids. In the boundary layer of a fluid flowing over a solid wall. Ihe shear stress varies with distance from Ihe wall bul ii may be considered nearly constant within a small fraction of the layer thickness. The concept is of particular importance in turbulent flow where it leads lo a theoretical derivation of the law of ihe wall," the logarithmic distribution of mean velocity. The constant stress layer is ihe best-known example of the equilibrium flow s near a wall. 

Non-Newtonian Viscosity In the cone-and-plate and parallel-disk torsional flow rheometer shown in Fig. 3.1, parts la and 2a, the experimentally obtained torque, and thus the % 2 component of the shear stress, are related to the shear rate y = y12 as follows for Newtonian fluids T12 oc y, implying a constant viscosity, and in fact we know from Newton s law that T12 = —/ . For polymer melts, however, T12 oc yn, where n < 1, which implies a decreasing shear viscosity with increasing shear rate. Such materials are called pseudoplastic, or more descriptively, shear thinning Defining a non-Newtonian viscosity,2 t],... 

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